Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 65

Chapter 4, Problem 65

Graph each rational function. ƒ(x)=4/(x-1)

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Identify the rational function given: $f(x) = \frac{4}{x - 1}$. Notice that the denominator $x - 1$ cannot be zero, so $x \neq 1$.

Determine the vertical asymptote by setting the denominator equal to zero: solve $x - 1 = 0$, which gives $x = 1$. This means the graph will have a vertical asymptote at $x = 1$.

Find the horizontal asymptote by analyzing the behavior of $f(x)$ as $x$ approaches infinity or negative infinity. Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is $y = 0$.

Calculate a few key points by substituting values of $x$ into the function, such as $x = 0$, $x = 2$, and $x = -1$, to understand the shape and position of the graph relative to the asymptotes.

Sketch the graph using the vertical asymptote at $x = 1$, the horizontal asymptote at $y = 0$, and the plotted points. Remember the graph will approach but never touch the asymptotes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined points. Graphing involves analyzing asymptotes, intercepts, and behavior near discontinuities.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. For f(x) = 4/(x-1), the vertical asymptote is at x = 1, indicating the graph will never cross this line but will approach it closely.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches infinity or negative infinity. For f(x) = 4/(x-1), since the degree of the numerator is less than the denominator, the horizontal asymptote is y = 0, meaning the graph approaches the x-axis at extreme values.

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The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x³ - 2x² - x+2. Use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of ƒ(x). ƒ (-2)

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Textbook Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5.

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